Rigidity via Modular Properties of Theta Functions

Abstract

In this paper we use methods of Liu to show that the twisted Dirac operators $D$ on certain bundles $\Phi$ considered by Guan and Wang are rigid. To do so, we use a Lefschetz formula and Atiyah-Bott localization to obtain formulas for the Lefschetz numbers $L$ of these operators $D$ in terms of Jacobi theta functions; then, using the translational and modular transformation properties of theta functions and the properties of their zeros, we prove $L$ is constant provided certain conditions on characteristic classes hold, thus showing the rigidity of $D$ on $\Phi$ under these conditions.